OR1-4
液滴浮遊法における内部対流が結晶粒成長に及ぼす影響
Effects of internal convection on grain growth structurein levitated droplets
谷本滉平 1, 白鳥英 2
Kohei TANIMOTO1 and Suguru SHIRATORI2
121

東京都市大学大学院，Graduate School, Tokyo City University,
東京都市大学，Tokyo City University,

Introduction
In metal additive manufacturing techniques, adding heterogeneous nucleation site particles is known toimprove grain structures, resulting in the strength of manufactured objects. The Hetero-3D project is being
carried out in an electrostatic levitation furnace (ELF) in the International Space Station (ISS) to investigate therefinement effects of heterogeneous nucleation sites on microstructures. Although the experiment in the ISS
allows us to extract the effect of refinement in pure form due to the absence of the buoyant convection in thedroplet, opportunities are limited because of the cost of the ISS mission. Thus, ground experiments are also
carried out through electrostatic levitation (ESL), aerodynamic levitation (ADL), and electromagneticlevitation (EML). In these levitation methods, the intensity of the droplet internal convections may besignificantly different, and it may affect the grain structures. This study aims to clarify the effect of convection
on crystal grain structures and, consequently, to extract the effect of adding heterogeneous nucleation sites ongrain refinement.
2Methods of investigation
2.1

Multi-phase-field model

In this study, a multi-phase-field (MPF) model is applied in numerical simulation to describe themicrostructure evolution of multiphase and polycrystalline materials. In the MPF model, the sum constraint
is considered for the phase fields 𝜙𝜙 as𝑛𝑛(1)� 𝜙𝜙𝑖𝑖 = 1,𝑖𝑖=1

where 𝑛𝑛 is the number of grains locally present. The dynamics of the interface are derived from thegeneralized free energy 𝐹𝐹, which is integration of sum of the double well potential 𝑓𝑓doub , the gradient energydensity 𝑓𝑓grad , and the chemical energy density 𝑓𝑓chem as
⬚

⬚

𝑛𝑛

𝑛𝑛

𝑛𝑛

𝑛𝑛

𝐹𝐹 = � �𝑓𝑓doub + 𝑓𝑓grad + 𝑓𝑓chem �𝑑𝑑𝑑𝑑 = � �� � �𝑊𝑊𝑖𝑖𝑖𝑖 𝜙𝜙𝑖𝑖 𝜙𝜙𝑗𝑗 � + � � �−𝑉𝑉𝑉𝑉

𝑖𝑖=1 𝑗𝑗=𝑖𝑖+1

𝑖𝑖=1 𝑗𝑗=𝑖𝑖+12𝑎𝑎𝑖𝑖𝑖𝑖∇𝜙𝜙𝑖𝑖 ∙ 𝜙𝜙𝑗𝑗 � + 𝑓𝑓chem � 𝑑𝑑𝑑𝑑.
2(2)Based on the Gibbs-free energy, the time evolution equation of 𝜙𝜙 is formulated as follows:

1 of 3𝑛𝑛

𝜕𝜕𝜙𝜙𝑖𝑖21 2
𝜕𝜕𝑓𝑓chem 𝜕𝜕𝑓𝑓chem
𝜙𝜙2= − � 𝑀𝑀𝑖𝑖𝑖𝑖 ���𝑊𝑊𝑖𝑖𝑖𝑖 − 𝑊𝑊𝑗𝑗𝑗𝑗 �𝜙𝜙𝑘𝑘 + �𝑎𝑎𝑖𝑖𝑖𝑖− 𝑎𝑎𝑗𝑗𝑗𝑗
�∇2 𝜙𝜙𝑘𝑘 � + �−�� ，𝜕𝜕𝜕𝜕
𝜕𝜕𝜙𝜙𝑖𝑖𝜕𝜕𝜙𝜙𝑗𝑗𝑛𝑛2(3)

𝑗𝑗=1

where 𝑀𝑀, 𝑊𝑊,and 𝑎𝑎 denote phase-field mobility, energy barrier, and gradient coefficient, respectively. For
quantitative simulation, these coefficients are determined from physical properties of materials. The chemical
driving force during the solidification process in pure substances is defined as the difference in free energy
density between the solid and liquid phases:
Δ𝑓𝑓 = −(𝑓𝑓𝑆𝑆 − 𝑓𝑓𝐿𝐿 ) = −𝐿𝐿(𝑇𝑇 − 𝑇𝑇𝑚𝑚 )
,𝑇𝑇𝑚𝑚

(4)

where 𝑓𝑓𝑆𝑆 , 𝑓𝑓𝐿𝐿 , 𝐿𝐿, and 𝑇𝑇𝑚𝑚 represent free energy of solid phase and liquid phase, latent heat, and equilibrium
temperature, respectively. Binary alloys are considered by calculating the Gibbs free energy from a linearizedphase diagram.

withΔ𝑓𝑓𝑖𝑖𝑖𝑖 = −𝑆𝑆𝑖𝑖𝑖𝑖 Δ𝑇𝑇𝑖𝑖𝑖𝑖 = −𝑆𝑆𝑖𝑖𝑖𝑖 �𝑇𝑇 − 𝑇𝑇𝑖𝑖𝑖𝑖 �,(5)

𝑇𝑇𝑖𝑖𝑖𝑖 = 𝑚𝑚𝑖𝑖:𝑖𝑖𝑖𝑖 (𝑐𝑐𝑖𝑖 − 𝑐𝑐𝑖𝑖𝑟𝑟 ) + 𝑇𝑇 𝑟𝑟 = 𝑚𝑚𝑖𝑖:𝑖𝑖𝑖𝑖 �𝑐𝑐𝑗𝑗 − 𝑐𝑐𝑗𝑗𝑟𝑟 � + 𝑇𝑇 𝑟𝑟 .

(6)Here 𝑆𝑆 denotes the entropy of fusion, 𝑚𝑚𝑖𝑖:𝑖𝑖𝑖𝑖 and 𝑚𝑚𝑖𝑖:𝑖𝑖𝑖𝑖 the gradient of phase lines, 𝑐𝑐𝑖𝑖 and 𝑐𝑐𝑗𝑗 theconcentration, 𝑐𝑐𝑖𝑖𝑟𝑟 and 𝑐𝑐𝑗𝑗𝑟𝑟 the reference concentration, 𝑇𝑇 𝑟𝑟 the reference temperature.
2.2

Numerical methods and implementation

A series of numerical simulations are carried out based on the MPF methods using the Academic versionof the OpenPhase. Pure titanium and the titanium alloy Ti-Al, which is often used in additive manufacturing,were considered as materials to be simulated. The time evolution of the orientation fields and concentration
field are coupled with the standard MPF model. One or several nuclei are generated within the computational
domain at the initial state. These nuclei are then grown according to the driving force of the Gibbs energy. The
convections are solved by the Lattice Boltzmann method, and the concentration and temperature fields areadvected by the flow.
3Results

In this study, simulations were carried out in 2-dimensional domain by changing flow velocities listed inTable 1. These velocity values are drawn from results of Usui et.al.1) Figures 1 shows the growth of single Ti.
In these cases, heat conduction is much faster than the convection heat transfer, so the temperature field isalmost not affected by the convection. Thus, the effect of the convection on the final grain structures isvanishingly small.
Table 1. The magnitude of internal convective flow velocity in droplets during the levitation methods
Levitation methods
Velocity order [m/s]

No convection

ESL𝑣𝑣 ≈ 1.0 × 10

EML condition

Figure 1. Pure titanium solidification

ADL−3𝑣𝑣 ≈ 1.0 × 10

No convection

EML−1𝑣𝑣 ≈ 1.0 × 100

ESL condition

Figure 2. Titanium alloy Ti-30at%Al solidification
2 of 3

No convection

ESL condition

ADL convection

EML condition

Figure 3. Multi-dendrite simulation of TiAl alloy with flow

10.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0Figure 4. The elliptical fitting

No coonvection

ESL condition

ADL condition

EML condition

Figure 5. Aspect ratio of grains

Figures 2 shows the dendrite growth of Ti-30at%Al alloy. In these cases, the concentration gradientbecomes steep at the grain boundary faced orthogonal to flow, leading to faster solid growth. Figure 3 shows
the growth of multiple grains of Ti-30at%Al dendrites simulated in 2D. To investigate the effect of convection
on the grain structures, grain interfaces are fitted by ellipses, as shown in Fig.4. The shape of the structures isevaluated as the aspect ratio, which is defined as the ratio of the minor to the major axes. Figure 5 shows theaspect ratios of the five grains in the computational domain and their average for each levitation method
condition. The average aspect ratio of the grain generated under ADL conditions was 1.25 times larger thanthat for the case of no convection. The tendency of the aspect ratio can be explained by the gradient of theconcentration field due to the mass transport by the convection.
4Conclusion

The effect of liquid convection on grain shape was elucidated through MPF simulations. The results ofthe present study showed that the final grain structures may be affected by the droplet internal convection in
the levitation methods.
References
1)

2)

T. Usui, S. Shiratori, K. Tanimoto, S. Ozawa, T. Ishikawa, S. Suzuki, H. Nagano, and K. Shimano: Surrogate models
for the magnitude of convection in droplets levitated through EML, ADL, and ESL methods. Int. J. Microgravity Sci.
Appl., 40 (2023) 400302, DOI: 10.15011/jasma.40.400302.
T. Takaki & A. Yamanaka: Phase-field method -Microstructure design through numerical simulation, 1st ed.,Yokendo (2012).
© 2024 by the authors. Submitted for possible open access publication under theterms and conditions of the Creative Commons Attribution (CC BY) license(http://creativecommons.org/li censes/by/4.0/).
3 of 3